
Such a distribution \(\mathcal {D}_{F(x,y,z,y^{\prime },y^{\prime \prime })}\)  is said to be in Monge normal form (see page 90 of [1]}).
When \(F(x,y,z,p,q)=q^2\) , we obtain the 1-forms of the Hilbert-Cartan distribution. In Section 5 of [2]}, it is shown how to associate canonically to such a \((2,3,5)\) -distribution a conformal class of metrics of split signature \((2,3)\)  (henceforth known as Nurowski's conformal structure or Nurowski's conformal metrics) such that the rank 2 distribution is isotropic with respect to any metric in the conformal class. The method of equivalence [3]} (also see the introduction to [4]}, Section 5 of [2]} and [6]}) produces the 1-forms \((\theta _1, \theta _2,\theta _3, \theta _4, \theta _5)\)  that gives a coframing for Nurowski's metric. These 1-forms satisfy the structure equations
\({\rm d}\theta _1&=\theta _1\wedge (2\Omega _1+\Omega _4)+\theta _2\wedge \Omega _2+\theta _3 \wedge \theta _4,\nonumber \\{\rm d}\theta _2&=\theta _1\wedge \Omega _3+\theta _2\wedge (\Omega _1+2\Omega _4)+\theta _3 \wedge \theta _5,\nonumber \\{\rm d}\theta _3&=\theta _1\wedge \Omega _5+\theta _2\wedge \Omega _6+\theta _3\wedge (\Omega _1+\Omega _4)+\theta _4 \wedge \theta _5,\\{\rm d}\theta _4&=\theta _1\wedge \Omega _7+\frac{4}{3}\theta _3\wedge \Omega _6+\theta _4\wedge \Omega _1+\theta _5 \wedge \Omega _2,\nonumber \\{\rm d}\theta _5&=\theta _2\wedge \Omega _7-\frac{4}{3}\theta _3\wedge \Omega _5+\theta _4\wedge \Omega _3+\theta _5\wedge \Omega _4,\nonumber \) 
